The $\mathrm{SO}(5)$ Deconfined Phase Transition under the Fuzzy Sphere Microscope: Approximate Conformal Symmetry, Pseudo-Criticality, and Operator Spectrum

The deconfined quantum critical point (DQCP) is an example of phase transitions beyond the Landau symmetry breaking paradigm that attracts wide interest. However, its nature has not been settled after decades of study. In this paper, we apply the recently proposed fuzzy sphere regularization to study the $\mathrm{SO}(5)$ non-linear sigma model (NL$\sigma$M) with a topological Wess-Zumino-Witten term, which serves as a dual description of the DQCP with an exact $\mathrm{SO}(5)$ symmetry. We demonstrate that the fuzzy sphere functions as a powerful microscope, magnifying and revealing a wealth of crucial information about the DQCP, ultimately paving the way towards its final answer. In particular, through exact diagonalization, we provide clear evidence that the DQCP exhibits approximate conformal symmetry. The evidence includes the existence of a conserved $\mathrm{SO}(5)$ symmetry current, a stress tensor, and integer-spaced levels between conformal primaries and their descendants. Most remarkably, we have identified 23 primaries and 76 conformal descendants. Furthermore, by examining the renormalization group flow of the lowest symmetry singlet as well as other primaries, we provide numerical evidence in favour of DQCP being pseudo-critical, with the approximate conformal symmetry plausibly emerging from nearby complex fixed points. The primary spectrum we compute also has important implications, including the conclusion that the $\mathrm{SO}(5)$ DQCP cannot describe a direct transition from the N\'eel to valence bond solid phase on the honeycomb lattice.